Here \({\mathcal {H}}_K\)  is the Reproducing Kernel Hilbert Space corresponding to the Gaussian kernel \(K({\mathbf {x}},{\mathbf {z}})=\exp (-\Vert {\mathbf {x}}-{\mathbf {z}}\Vert ^2)\)  and \({\mathcal {S}}\subset {\mathcal {H}}_K\)  is the manifold of interpolating functions in \({\mathcal {H}}_K\) .
Note that \(f_{\rm ker}({\mathbf {x}})\)  defined here is the same function defined in \(Eq.~\ref {eq:kernel}\) . This equality is known as the Representer Theorem [1]}, [2]}.
